부분합 점화식과 부분분수를 이용한 시그마

$S _{n} = 2 a _{n} - pn$에서 $S _{1} = a _{1} = 2 a _{1} - p$ $\therefore$ $a _{1} = p$ $S _{n} - S _{n - 1} = 2 \left( a _{n} - a _{n - 1} \right) - pn + p \left( n - 1 \right) \left( n \geq 2 \right)$ $\therefore$ $a _{n} = 2 a _{n - 1} + p \left( n \geq 2 \right)$ $a _{n} = 2 a _{n - 1} + p$에서 $a _{n} + p$$= 2 \left( a _{n - 1} + p \right)$이므로 $a _{n} + p$$= \left( a _{1} + p \right) \times 2 ^{n - 1}$ $= 2 p \times 2 ^{n - 1}$ $\therefore$ $a _{n} = p \left( 2 ^{n} - 1 \right) \left( n \geq 2 \right)$ $S _{1} = a _{1}$이므로 $a _{n} = p \left( 2 ^{n} - 1 \right) \left( n \geq 1 \right)$ $\displaystyle \sum\limits _{k = 1} ^{6} \frac{p + a _{k}}{a _{k} a _{k + 1}}$$\displaystyle = \sum\limits _{k = 1} ^{6} {\frac{p + a _{k}}{a _{k} \left( 2 a _{k} + p \right)}}$ $\displaystyle = \sum\limits _{k = 1} ^{6} \left( \frac{1}{a _{k}} - \frac{1}{2 a _{k} + p} \right)$ $\displaystyle = \sum\limits _{k = 1} ^{6} \left( \frac{1}{a _{k}} - \frac{1}{a _{k + 1}} \right)$ $\displaystyle = \left( \frac{1}{a _{1}} - \frac{1}{a _{2}} \right) + \left( \frac{1}{a _{2}} - \frac{1}{a _{3}} \right) + \cdots + \left( \frac{1}{a _{6}} - \frac{1}{a _{7}} \right)$ $\displaystyle = \frac{1}{a _{1}} - \frac{1}{a _{7}}$ $\displaystyle = \frac{1}{p} - \frac{1}{127 p}$ $\displaystyle = \frac{126}{127 p}$ 따라서 $\displaystyle \frac{126}{127 p} = 3$이므로 $\displaystyle p = \frac{42}{127}$